#7628307

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,28)(29,32);(10,14)(13,16)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31);(19,26)(22,30)(25,27)(29,31);(18,24)(22,30)(23,32)(29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=h,c^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535503);